I'm posting an update since I've been dealing with this annoying chirp for the better part of 2 years. It worsened this summer so I started aggressively trouble shooting. I noticed that the chirp was eliminated when I kept light pressure on the brake peddle, confirming it was brake related vs. suspension. I decided to pull the pads to inspect and found the backing on my Axxis Deluxe pads with 20k miles had completely separated from the backing plate. The Axxis pads use a cheap, fiberglass/plastic sticker as an anti-vibration shim. It sucks. Not only were the pads chirping with movement over sharp bumps, they repeatedly squealed even after reinstalling them twice within the first couple of months.
I'm not looking to start a debate re: Axxis but I personally won't use them again. I bought a set of Pagid OEM pads off Autohaus and the chirp is 100% gone. Better yet, I have the excellent cold bite and factory-like brake performance back which I've missed for 2 years. The dust level isn't too bad, either. One noticeable difference is the backing shim on the Pagids which is a rubber vs. the cheap sticker used on the Axxis.
If it sounds like you have a bird in your suspension, check the brake pads.